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Abstract The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking. 

This paper proposes an approach for the estimation of a time-varying Hurst exponent to allow accurate identification of multifractional Brownian motion (MFBM). The contribution provides a prescription for how to deal with the MFBM measurement data to solve regression and classification problems. Theoretical studies are supplemented with computer simulations and real-world examples. Those prove that the procedure proposed in this paper outperforms the best-in-class algorithm. 

Abstract Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion processes in complex systems are usually not stationary, the traditional Wiener-Khinchin theorem for the analysis of power spectral densities is invalid. Here, we employ a recently developed tool named aging Wiener-Khinchin theorem to derive the power spectral density of fractional Brownian motion coexisting with a scale-free continuous time random walk, the two most typical anomalous diffusion processes. Using this analysis, we characterize the motion of voltage-gated sodium channels on the surface of hippocampal neurons. Our results show aging where the power spectral density can either increase or decrease with observation time depending on the specific parameters of both underlying processes. 

{"Abstract":["Datasets generated in the report "Aging power spectrum of membrane protein transport and other subordinated random walks". Included data are:<\/p>\n\nNumerical simulations <\/strong>\nRWdata1.mat: 10,000 realizations, subordinated random walk with Hurst exponent, H<\/em>=0.3 and \\(\\alpha\\)=0.4.\nRWdata3.mat: 10,000 realizations, subordinated random walk with Hurst exponent, H<\/em>=0.7 and \\(\\alpha\\)=0.4.\nRWdata8.mat: 5,000 realizations, subordinated random walk with Hurst exponent, H<\/em>=0.75 and \\(\\alpha\\)=0.8.\nRWdataCTRW.mat: 10,000 realizations, continuous time random walk (CTRW), \\(\\alpha\\)=0.7.<\/p>\n\nSpectra of simulations<\/strong>\nPSDdata1.mat: Power spectral density (PSD) of a subordinated random walk with Hurst exponent, H<\/em>=0.3 and \\(\\alpha\\)=0.4. Five different realization times are used to compute the PDS: 2^8, 2^10, 2^12, 2^14, and 2^16.\nPSDdata3.mat: PSD of a subordinated random walk with Hurst exponent, H<\/em>=0.7 and \\(\\alpha\\)=0.4. Five different realization times are used to compute the PDS: 2^8, 2^10, 2^12, 2^14, and 2^16.\nPSDdata8.mat: PSD of a subordinated random walk with Hurst exponent, H<\/em>=0.75 and \\(\\alpha\\)=0.8. Four different realization times are used to compute the PDS: 2^15, 2^16, 2^17, and 2^18.\nPSDs_CTRW.mat: PSD of a continuous-time random walk (CTRW), \\(\\alpha\\)=0.7. Five different realization times are used to compute the PDS: 2^8, 2^10, 2^12, 2^14, and 2^16.<\/p>\n\nExperimental data of Nav1.6 channels in the soma of hippocampal neurons<\/strong>\nNavMSDtimes.csv: ensemble-averaged (EA) MSD and time-averaged (TA) MSD. The TA-MSD is measured for three observation times, 64, 128, and 256 frames (3.2, 6.4, and 12.8 s).\nNavPSD.csv: Power spectral density (PSD) measured for three observation times, 64, 128, and 256 frames.<\/p>"],"Other":["We acknowledge the support of the National Science Foundation grant 2102832 (to DK) and Israel Science Foundation grant 1898/17 (to EB).","{"references": ["Fox, Z.R., Barkai, E. & Krapf, D. Aging power spectrum of membrane protein transport and other subordinated random walks. Nat Commun 12, 6162 (2021). https://doi.org/10.1038/s41467-021-26465-8"]}"]} 

Abstract The resolution of fluorescence microscopy images is limited by the physical properties of light. In the last decade, numerous super-resolution microscopy (SRM) approaches have been proposed to deal with such hindrance. Here we present Mean-Shift Super Resolution (MSSR), a new SRM algorithm based on the Mean Shift theory, which extends spatial resolution of single fluorescence images beyond the diffraction limit of light. MSSR works on low and high fluorophore densities, is not limited by the architecture of the optical setup and is applicable to single images as well as temporal series. The theoretical limit of spatial resolution, based on optimized real-world imaging conditions and analysis of temporal image stacks, has been measured to be 40 nm. Furthermore, MSSR has denoising capabilities that outperform other SRM approaches. Along with its wide accessibility, MSSR is a powerful, flexible, and generic tool for multidimensional and live cell imaging applications.